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Hometopic entry on axioms and foundations of homology and cohomology theories

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# topic entry on axioms and foundations of homology and cohomology theories

# 1 Axioms for Homology and Cohomology theories

1. Axioms for homology theory and uniqueness theorems

2. Cech types

3. $K$-theory

4. Generalized cohomology

5. Generalized (extraordinary) homology and cohomology theories

6. 7. (Co)homology of commutative rings and algebras (e.g., Hochschild, André–Quillen, cyclic, dihedral, etc.)

8. 9. Homology with local coefficients, equivariant cohomology

10. 11. Cohomology in Noncommutative algebraic geometry

12. Classifying spaces for foliations; Gelfand-Fuks cohomology

13. Intersection homology and cohomology

14. Elliptic cohomology

15. Equivariant homology and cohomology

16. Homology and homotopy of topological groups and related structures

17. Homotopy Quantum Field Theories and Axiomatic Quantum Field Theories

18. Non-Abelian Homological Algebra

19. Grothendieck’s ‘Anabelian Geometry’

20. Other homology theories–Your new additions

# References

- 1
Hatcher, A. 2001.
*Algebraic Topology*(textbook on line)., Cambridge University Press; Cambridge, UK., 405 pages. - 2 Ronald Brown: Topology and Groupoids, BookSurge LLC (2006).
- 3
Ronald Brown R, P.J. Higgins, and R. Sivera.:
*“Non-Abelian algebraic topology”*, (2008). - 4 R. Brown and J.-L. Loday: Homotopical excision, and Hurewicz theorems, for n-cubes of spaces, Proc. London Math. Soc., 54:(3), 176-192, (1987).
- 5 R. Brown and J.-L. Loday: Van Kampen Theorems for diagrams of spaces, Topology, 26: 311-337 (1987).
- 6 R. Brown and C.B. Spencer: Double groupoids and crossed modules, Cahiers Top. Géom. Diff., 17 (1976), 343-362.
- 7
Allain Connes:
*Noncommutative Geometry*, Academic Press 1994. - 8
May, J.P. 1999,
*A Concise Course in Algebraic Topology.*, The University of Chicago Press: Chicago

## Mathematics Subject Classification

11E72*no label found*55N99

*no label found*55N25

*no label found*55N30

*no label found*55N05

*no label found*12G05

*no label found*14A22

*no label found*11R34

*no label found*55N40

*no label found*

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